`x - 4y + 3z - 2w = 9, 3x - 2y + z - 4w = -13, ` `-4x + 3y - 2z + w = -4, -2x + y - 4z + 3w = -10` Use matricies to solve the system of equations. Use Gaussian elimination with back-substitution.

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The above system of equations can be represented by the coefficient matrix A and right hand side matrix b as follows:



The augmented matrix can be written as,


Now lets, perform the various row operations to bring the above matrix in the row-echelon form,

Rewrite the 2nd Row `(R_2)` as `(R_2-3R_1)`


Rewrite the 3rd Row`(R_3)` as`(R_3+4R_1)` 


Rewrite the 4th Row`(R_4)` as`(R_4+2R_1)`  


Rewrite the 2nd Row`(R_2)` as`(2(R_2+R_3)-R_4)`


Rewrite the 3rd Row`(R_3)` as`(R_3+13R_2)`


Rewrite the 4th Row`(R_4)` as `(R_4+7R_2)`  


Rewrite the 3rd Row`(R_3)` as `(R_3-R_4)`


Rewrite the 3rd Row by dividing it with 20,


Rewrite the 4th Row by dividing it with 16,


Rewrite the 4th Row as `(R_3-R_4)`


Now the matrix is in row-echelon form, and we can perform the back substitution on the corresponding system,

`x-4y+3z-2w=9`     -----  Eq:1

`y+2z-9w=-24`        -----  Eq:2

`z-3w=-6`                 -----  Eq:3


Substitute back the value of w in Eq:3,





Substitute back the value of w and z in Eq:2,





Substitute back the value of w,z and y in Eq:1,






So the solutions are x=-1,y=0,z=6 and w=4

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